Quasiparticle calculation of the dielectric response of silicon and germanium

We calculate the dielectric function ${\mathrm{\epsilon }}_1$(ω) of silicon and germanium for frequencies below the direct band gap at the experimental lattice constant as well as the pressure dependence of the static dielectric constant ${\mathrm{\epsilon }}_1$(P). Our theory is an extension of the local-density approximation (LDA). We include self-energy effects by adding an operator to the usual LDA Hamiltonian; i.e., ${\mathit{H}}_{\mathbf{k}}$=${\mathit{H}}_{\mathbf{k}}^{\mathrm{LDA}}$+${\mathrm{\Delta }}_{\mathbf{k}}$ ${\mathit{P}}_{\mathit{c}\mathbf{k}}$. This form leads to a Ward-identity replacement p→(${\mathrm{\varepsilon }}_{\mathit{n}\mathbf{k}}$-${\mathit{H}}_{\mathbf{k}}^{\mathrm{LDA}}$ ${)}^{\mathrm{-}1}$(${\mathrm{\varepsilon }}_{\mathit{n}\mathbf{k}}$ -${\mathit{H}}_{\mathbf{k}}$)p. Our theory is in agreement with experiment for ${\mathrm{\epsilon }}_1$(ω) at the level of a few percent. For ${\mathrm{\epsilon }}_1$(P), we are in agreement with experiments for silicon and some of the conflicting experiments for germanium. A prediction is made for the pressure dependence of the dielectric function of silicon beyond the linear regime.